SubeV Hubble scale inflation within gauge mediated supersymmetry breaking
Abstract
Minimal Supersymmetric Standard Model with gauge mediated supersymmetry breaking has all the necessary ingredients for a successful subeV Hubble scale inflation eV. The model generates the right amplitude for scalar density perturbations and a spectral tilt within the range, . The reheat temperature is TeV, which strongly prefers electroweak baryogenesis and creates the right abundance of gravitinos with a mass keV to be the dark matter.
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Embedding a very low scale inflation within a particle physics model is a challenging problem. It is not only difficult to obtain sufficient number of efoldings, right amplitude for the scalar density perturbations, the right tilt in the power spectrum, but also generating baryon asymmetry and dark matter simultaneously.
Recently there has been a real progress in our understanding of embedding inflation within particle physics, particularly within the Minimal Supersymmetric Standard Model (MSSM), where the inflaton belongs to the MSSM instead of being an adhoc gauge singlet. Foremost the models AEGM ; AKM not only predict the right amplitude of the scalar density perturbations and the tilted spectrum, see also AM , but are also testable at LHC AEGJM ^{1}^{1}1The inflaton candidates are, AEGM and AKM flat directions. Here denotes lefthanded sleptons, denote righthanded squarks and sleptons, respectively, denotes righthanded sneutrinos and is the Higgs which gives masses to the up type quarks. The mass should be GeV AEGM , which is within the reach of LHC. For other attempts of inflation with flat directions, see FEW ; LYTH1 . For a review on MSSM flat directions, see MSSMREV ..
In this paper we provide a simple example of a subeV Hubble scale inflation, eV, embedded within MSSM. This is realizable provided supersymmetry (SUSY) breaking is communicated via gauge mediation, i.e. gauge mediated supersymmetry breaking (GMSB) GMSB , in contrast to Refs. AEGM ; AKM where we assumed gravity mediation. We shall predict the reheat temperature around TeV, which strongly favors electroweak baryogenesis within MSSM and gravitino as a dark matter candidate. Thus all the ingredients for a successful cosmology are naturally contained within MSSM.
Let us first highlight relevant points of the model:

Within GMSB the gravitino is the lightest SUSY particle (LSP). Therefore a reheat temperature of will lead to a sufficient relic abundance for the (stable) gravitino as a dark matter candidate STEFFEN .
Let us now consider an MSSM flat direction, , lifted by a nonrenormalizable () superpotential term (for a detailed dynamics on multiple flat direction, see EJM ):
(1) 
where is the superfield which contains the flat direction . Within MSSM all the flat directions are lifted by operators DRT ; GKM . The cutoff scale is , therefore the above superpotential is a reflection of integrating out the physics above the GUT scale, and we assume the nonrenormalizable coupling to be . Note that the new physics does not necessarily have to be tied to the GUT physics. For example a gauged may appear at an intermediate scale . This will amount to the same superpotential as in Eq. (1) parameterized by , so long as . This is conceivable as is a product of the Yukawa couplings associated with the new interaction terms beyond the MSSM.
In GMSB the twoloop correction to the flat direction potential results in a logarithmic term above the messenger scale, i.e. dGMM . Together with the term this leads to the scalar potential
(2) 
where and TeV is the soft SUSY breaking mass at the weak scale. For , usually the gravity mediated contribution, , dominates the potential where is the gravitino mass. Here we will concentrate on the VEVs .
In Eq. (2), and denote the radial and the angular coordinates of the complex scalar field respectively, while is the phase of term (thus is a positive quantity with a dimension of mass). Note that the first and third terms in Eq. (2) are positive definite, while the term leads to a negative contribution along the directions where . The cosmological importance of an term can be found in MSSMREV ; DRT ; MARIEKE ; CURVATON .
Although individual terms are unable to support a subPlanckian VEV inflation, but as shown in Refs. AEGM ; AKM ; AEGJM ; AM , a successful inflation can be obtained near the saddle point, which we find by solving, (where derivative is w.r.t ).
(3)  
(4) 
In the vicinity of the saddle point, we obtain the total energy density and the third derivative of the potential to be:
(5)  
(6) 
There are couple of interesting points, first of all note that the scale of inflation is extremely low in our case, barring some small coefficients of order one, the Hubble scale during inflation is given by:
(7) 
for TeV. For such a low scale inflation usually it is extremely hard to obtain the right phenomenology. But there are obvious advantages of having a low scale inflation, . The supergravity corrections and the TransPlanckian corrections are all negligible AEGJM , therefore the model predictions are trustworthy.
Perturbations which are relevant for the COBE normalization are generated a number efoldings before the end of inflation. The value of depends on thermal history of the universe and the total energy density stored in the inflaton, which in our case is bounded by, . The required number of efoldings yields in our case, BURGESS , provided the universe thermalizes within one Hubble time. Although within SUSY thermalization time scale is typically very long AVERDI1 , however, in this particular case it is possible to obtain a rapid thermalization.
Near the vicinity of the saddle point, , the potential is extremely flat and one enters a regime of selfreproduction LINDE . The selfreproduction regime lasts as long as the quantum diffusion is stronger than the classical drag; , for , where . From then on, the evolution is governed by the classical slow roll. Inflation ends when , which happens at , where
(8) 
Assuming that the classical motion is due to the third derivative of the potential, , the total number of efoldings during the slow roll period is found to be:
(9) 
This simplifies to
(10) 
Let us now consider the adiabatic density perturbations. Despite eV, the flat direction can generate adequate density perturbations as required to explain to match the observations. Recall that inflation is driven by , we obtain
(11) 
Note that for TeV, and , we match the current observations WMAP3 , when GeV. The validity of Eq. (2) for such a large VEV requires that . For TeV this yields the bound on the gravitino mass, MeV, which is compatible with the dark matter constraints as we will see.
We can naturally satisfy Eq. (11) provided, . The nonrenormalizable operator, , points towards two MSSM flat directions out of many,
(12) 
As we discussed before in AEGM , these are the only directions which are suitable for inflation as they give rise to a nonvanishing term. Note that the inflatons are now the gauge invariant objects. The total number of efoldings, during the slow roll inflation, after using Eq. (10) yields,
(13) 
While the spectral tilt and the running of the power spectrum are determined by .
(14)  
(15) 
where . Note that the spectral tilt is slightly away from the result of the current WMAP 3 years data, on the other hand running of the spectrum is well inside the current bounds WMAP3 .
At first instance one would discard the model just from the slight mismatch in the spectral tilt from the current observations. However note that our analysis strictly assumes that the slow roll inflation is driven by . This is particularly correct if and . Let us then study the case when , as discussed in AM .
The latter case can be studied by parameterizing a small deviation from the exact saddle point condition by solving near the point of inflection, where we wish to solve and we get upto 1st order in the deviation, ,
(16) 
with and are the saddle point solutions. Then the 1st derivative is given by
(17) 
Therefore the slope of the potential is determined by, .
Note that both the terms on the righthand side are positive. The fact that can lead to an interesting changes from the saddle point behavior, for instance the total number of efoldings is now given by
(18) 
First of all note that by including , we are slightly away from the saddle point and rather close to the point of inflection. This affects the total number of efoldings during the slow roll. It is now much less than that of , i.e. , see Eq. (13).
When both the terms in the denominator of the integrand contributes equally then there exists an interesting window.
(19) 
where
(20) 
The lower limit in Eq. (19) is saturated when , while the upper limit is saturated when . It is also easy to check that there will be no selfreproduction regime for the field values determined by .
It is a straightforward but a tedious exercise to demonstrate that when the upper limit of Eq. (19) is saturated the spectral tilt becomes , when the lower limit is satisfied we recover the previous result with . This value, , can be easily understood as (where corresponds to the VeV where the end of inflation corresponds to ), in which case, . Therefore the spectral tilt becomes nearly scale invariant. We therefore find a range AM ,
(21) 
whose width is within the error of the central limit WMAP3 . Similarly the running of the spectral tilt gets modified too but remains within the observable limit ^{2}^{2}2A similar exercise can be done for the running of the spectral tilt and the running lies between AM ., while the amplitude of the power spectrum is least affected AM .
Let us now discuss the issue of reheating and thermalization. Important point is to realize that the inflaton belongs to the MSSM, i.e. and , both carry MSSM charges and both have gauge couplings to gauge bosons and gauginos. After inflation the condensate starts oscillating. The effective frequency of the inflaton oscillations in the Logarithmic potential, Eq. (2), is of the order of , while the expansion rate is given by . This means that within one Hubble time the inflaton oscillates nearly times. The motion of the inflaton is strictly one dimensional from the very beginning. During inflation, the imaginary direction is very heavy and settles down in the minimum of the potential.
An efficient bout of particle creation occurs when the inflaton crosses the origin, which happens twice in every oscillation. The reason is that the fields which are coupled to the inflaton are massless near the point of enhanced symmetry. Mainly electroweak gauge fields and gauginos are then created as they have the largest coupling to the flat direction. The production takes place in a short interval. Once the inflaton has passed by the origin, the gauge bosons/gauginos become heavy by virtue of VeV dependent masses and they eventually decay into particles sparticles, which creates the relativistic thermal bath. This is socalled instant preheating mechanism INSTANT . In a favorable condition, the flat direction VeV coupled very weakly to the flat direction inflaton could also enhance the perturbative decay rate of the inflaton ABM . In any case there is no nonthermal gravitino production MAROTO as the energy density stored in the inflaton oscillations is too low.
A full thermal equilibrium is reached when and are established AVERDI1 . The maximum temperature of the plasma is given by
(22) 
when the flat direction, either or evaporates completely. This naturally happens at the weak scale. There are two very important consequences which we summarize below.
Hot or cold electroweak Baryogenesis: The model strongly favors electroweak baryogenesis within MSSM. Note that the reheat temperature is sufficient enough for a thermal electroweak baryogenesis BARYOREV .
However, if the thermal electroweak baryogenesis is not triggered, then cold electroweak baryogenesis is still an option GKS . During the cold electroweak baryogenesis, the large gauge field fluctuations give rise to a nonthermal sphaleron transition. In our case it is possible to excite the gauge fields of during instant preheating provided the inflaton is . The as an inflaton carries the same quantum number which has a anomaly and large gauge field excitations can lead to nonthermal sphaleron transition to facilitate baryogenesis within MSSM.
Gravitino dark matter: Within GMSB gravitinos are the LSP and if the parity is conserved then they are an excellent candidate for the dark matter. There are various sources of gravitino production in the early universe STEFFEN ; AHJMP . However in our case the thermal production is the dominant one and mainly helicity gravitinos are created. Gravitinos thus produced have the correct dark matter abundance for dGMM ; BUCH
(23) 
where is the gluino mass. For keV, Eq. (23) is easily satisfied for TeV and TeV. We remind that for keV gravitinos produced from the sfermion decays overclose the universe dGMM .
Before concluding we should also highlight that the existence of a saddle point does not get spoiled through radiative corrections, see Ref. AEGJM . To summarize, we provided a truly low scale inflation model with eV, embedded within MSSM, provided GMSB is the correct paradigm. Although inflation occurs at such low scales, the model predictions match the current WMAP data and the reheat temperature of TeV is sufficient enough to trigger either hot or cold electroweak baryogenesis. The model also produces sufficient abundance of gravitinos to be the dark matter candidate. Thus inflation within GMSB connects the physics of microwave background radiation to a successful dark matter and a baryogenesis scenario whose ingredients are testable at the LHC.
We wish to thank Alex Kusenko and Misha Shaposhnikov for helpful discussion. The research of RA was supported by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported in part by the Government of Canada through NSERC and by the provine of Ontario through MEDT. The research of AM is partly supported by the European Union through Marie Curie Research and Training Network “UNIVERSENET” (MRTNCT2006035863).
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